### Abstract

Let G be a reductive algebraic group over the complex number filed, and K = G^{θ} be the fixed points of an involutive automorphism θ of G so that (G,K) is a symmetric pair. We take parabolic subgroups P and Q of G and K respectively and consider a product of partial flag varieties G/P and K/Q with diagonal Kaction. The double flag variety G/P × K/Q thus obtained is said to be of finite type if there are finitely many K-orbits on it. A triple flag variety G/P^{1} × G/P^{2} × G/P3 is a special case of our double flag varieties, and there are many interesting works on the triple flag varieties. In this paper, we study double flag varieties G/P × K/Q of finite type. We give efficient criterion under which the double flag variety is of finite type. The finiteness of orbits is strongly related to spherical actions of G or K. For example, we show a partial flag variety G/P is K-spherical if a product of partial flag varieties G/P ×G/θ(P) is G-spherical. We also give many examples of the double flag varieties of finite type, and for type AIII, we give a classification when P = B is a Borel subgroup of G.

Original language | English |
---|---|

Pages (from-to) | 79-99 |

Number of pages | 21 |

Journal | Journal of Lie Theory |

Volume | 21 |

Issue number | 1 |

Publication status | Published - 2011 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Journal of Lie Theory*,

*21*(1), 79-99.

**Double flag varieties for a symmetric pair and finiteness of orbits.** / Nishiyama, Kyo; Ochiai, Hiroyuki.

Research output: Contribution to journal › Article

*Journal of Lie Theory*, vol. 21, no. 1, pp. 79-99.

}

TY - JOUR

T1 - Double flag varieties for a symmetric pair and finiteness of orbits

AU - Nishiyama, Kyo

AU - Ochiai, Hiroyuki

PY - 2011

Y1 - 2011

N2 - Let G be a reductive algebraic group over the complex number filed, and K = Gθ be the fixed points of an involutive automorphism θ of G so that (G,K) is a symmetric pair. We take parabolic subgroups P and Q of G and K respectively and consider a product of partial flag varieties G/P and K/Q with diagonal Kaction. The double flag variety G/P × K/Q thus obtained is said to be of finite type if there are finitely many K-orbits on it. A triple flag variety G/P1 × G/P2 × G/P3 is a special case of our double flag varieties, and there are many interesting works on the triple flag varieties. In this paper, we study double flag varieties G/P × K/Q of finite type. We give efficient criterion under which the double flag variety is of finite type. The finiteness of orbits is strongly related to spherical actions of G or K. For example, we show a partial flag variety G/P is K-spherical if a product of partial flag varieties G/P ×G/θ(P) is G-spherical. We also give many examples of the double flag varieties of finite type, and for type AIII, we give a classification when P = B is a Borel subgroup of G.

AB - Let G be a reductive algebraic group over the complex number filed, and K = Gθ be the fixed points of an involutive automorphism θ of G so that (G,K) is a symmetric pair. We take parabolic subgroups P and Q of G and K respectively and consider a product of partial flag varieties G/P and K/Q with diagonal Kaction. The double flag variety G/P × K/Q thus obtained is said to be of finite type if there are finitely many K-orbits on it. A triple flag variety G/P1 × G/P2 × G/P3 is a special case of our double flag varieties, and there are many interesting works on the triple flag varieties. In this paper, we study double flag varieties G/P × K/Q of finite type. We give efficient criterion under which the double flag variety is of finite type. The finiteness of orbits is strongly related to spherical actions of G or K. For example, we show a partial flag variety G/P is K-spherical if a product of partial flag varieties G/P ×G/θ(P) is G-spherical. We also give many examples of the double flag varieties of finite type, and for type AIII, we give a classification when P = B is a Borel subgroup of G.

UR - http://www.scopus.com/inward/record.url?scp=78650864076&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78650864076&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:78650864076

VL - 21

SP - 79

EP - 99

JO - Journal of Lie Theory

JF - Journal of Lie Theory

SN - 0949-5932

IS - 1

ER -